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Properties

Label 4-117e2-1.1-c1e2-0-7
Degree 44
Conductor 1368913689
Sign 11
Analytic cond. 0.8728220.872822
Root an. cond. 0.9665650.966565
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 7-s + 5·13-s − 8·19-s − 10·25-s + 2·28-s − 14·31-s + 10·37-s + 13·43-s + 7·49-s + 10·52-s + 13·61-s − 8·64-s − 11·67-s + 34·73-s − 16·76-s − 26·79-s + 5·91-s − 5·97-s − 20·100-s − 26·103-s − 38·109-s + 11·121-s − 28·124-s + 127-s + 131-s − 8·133-s + ⋯
L(s)  = 1  + 4-s + 0.377·7-s + 1.38·13-s − 1.83·19-s − 2·25-s + 0.377·28-s − 2.51·31-s + 1.64·37-s + 1.98·43-s + 49-s + 1.38·52-s + 1.66·61-s − 64-s − 1.34·67-s + 3.97·73-s − 1.83·76-s − 2.92·79-s + 0.524·91-s − 0.507·97-s − 2·100-s − 2.56·103-s − 3.63·109-s + 121-s − 2.51·124-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + ⋯

Functional equation

Λ(s)=(13689s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(13689s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1368913689    =    34⋅1323^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 0.8728220.872822
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 13689, ( :1/2,1/2), 1)(4,\ 13689,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3273365501.327336550
L(12)L(\frac12) ≈\approx 1.3273365501.327336550
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
13C2C_2 1−5T+pT2 1 - 5 T + p T^{2}
good2C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−5T+pT2)(1+4T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C2C_2 (1+T+pT2)(1+7T+pT2) ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} )
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
31C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
37C2C_2 (1−11T+pT2)(1+T+pT2) ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} )
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C2C_2 (1−8T+pT2)(1−5T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C2C_2 (1−14T+pT2)(1+T+pT2) ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} )
67C2C_2 (1−5T+pT2)(1+16T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} )
71C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
73C2C_2 (1−17T+pT2)2 ( 1 - 17 T + p T^{2} )^{2}
79C2C_2 (1+13T+pT2)2 ( 1 + 13 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
97C2C_2 (1−14T+pT2)(1+19T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} )
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.68878978936650234687996714425, −13.29495144815058327557878512138, −12.71655998394923824990111930580, −12.30038021614666650701412544902, −11.46977469887847279077121922683, −11.21867273264163268591275018335, −10.82091562245990221907594748138, −10.42730372558088741522092921022, −9.359902462929009005981090779922, −9.175929646603444156515207466466, −8.197187863482219290864543145817, −7.969176601547087292910961275462, −7.12479528520181278145061038849, −6.63122969102480163207734998762, −5.81473512122248334257704625242, −5.65501082476237562525325714497, −4.06267316216209923172480215955, −4.02105013198232063605481217469, −2.52895160986709551773881672407, −1.80387006944753394195323284491, 1.80387006944753394195323284491, 2.52895160986709551773881672407, 4.02105013198232063605481217469, 4.06267316216209923172480215955, 5.65501082476237562525325714497, 5.81473512122248334257704625242, 6.63122969102480163207734998762, 7.12479528520181278145061038849, 7.969176601547087292910961275462, 8.197187863482219290864543145817, 9.175929646603444156515207466466, 9.359902462929009005981090779922, 10.42730372558088741522092921022, 10.82091562245990221907594748138, 11.21867273264163268591275018335, 11.46977469887847279077121922683, 12.30038021614666650701412544902, 12.71655998394923824990111930580, 13.29495144815058327557878512138, 13.68878978936650234687996714425

Graph of the ZZ-function along the critical line